We analyze the energy spectrum of graphene in the presence of spin-orbitcoupling and a unidirectionally periodic Zeeman field, focusing on thestability and location of Dirac points it may support. It is found that theDirac points at the $K$ and $K'$ points are generically moved to otherlocations in the Brillouin zone, but that they remain present when the Zeemanfield $\vec{\Delta}(x)$ integrates to zero within a unit cell. A large varietyof locations for the Dirac points is shown to be possible: when $\vec\Delta\parallel \hat{z}$ they are shifted from their original locations along thedirection perpendicular to the superlattice axis, while realizations of$\vec\Delta(x)$ that rotate periodically move the Dirac points to locationsthat can reflect the orbit of the rotating electron spin as it moves through aunit cell. When a uniform Zeeman field is applied in addition to a periodic$\vec\Delta \parallel \hat{z}$ integrating to zero, the system can be broughtinto a metallic, Dirac semimetal, or insulating state, depending on thedirection of the uniform field. The latter is shown to be an anomalous quantumHall insulator.
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